Introduction to Algebra

353 Pages·2008·6.05 MB·English

by  Peter J. Cameron

Introduction to Algebra
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Introduction to Algebra
Second Edition
Peter J. Cameron
Queen Mary, University of London
1
3
GreatClarendonStreet,Oxfordox26dp
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Preface
This new edition of my algebra textbook has a number of changes.
The most significant is that the book now tries to live up to its title better
than it did in the previous edition: the introductory chapter has more than
doubledinlength,includingbasicmaterialonproofs,numbers,algebraicmanip-
ulations, sets, functions, relations, matrices, and permutations. I hope that it is
now accessible to a first-year mathematics undergraduate, and suitable for use
in a first-year mathematics course. Indeed, much of this material comes from
a course (also with the title ‘Introduction to Algebra’) which I gave at Queen
Mary, University of London, in spring 2007.
I have also revised and corrected the rest of the book, while keeping the
structure intact. In particular, the pace of the first chapter is quite gentle; in
Chapters 2 and 3 it picks up a bit, and in the later chapters it is a bit faster
again. Once you are used to the way I write mathematics, you should be able
to take this in your stride. Since the book is intended to be used in a variety of
courses,thereisacertainamountofrepetition.Forexample,conceptsorresults
introduced in exercises may be dealt with later in the main text. New material
ontheAxiomofChoice,p-groups,andlocalringshasbeenadded,andthereare
many new exercises.
Iamgratefultomanypeoplewhohavehelpedme.Firstandforemost,Robin
Chapman,forspottingmanymisprintsandmakingmanysuggestions;andCsaba
Szabo´, who encouraged his students (named below) to proofread the book very
thoroughly! Also, Gary McGuire spotted a gap in the proof of the Fundamental
TheoremofGaloisTheory,andR.A.BaileysuggestedadifferentproofofSylow’s
Theorem. The people who notified me of errors in the book, or who suggested
improvements (as well as the above) are Laura Alexander, Richard Anderson,
M. Q. Baig, Steve DiMauro, Karl Fedje, Emily Ford, Roderick Foreman, Will
Funk, Rippon Gupta, Matt Harvey, Jessica Hubbs, Young-Han Kim, Bill Mar-
tin, William H. Millerd, Ioannis Pantelidakis, Brandon Peden, Nayim Rashid,
Elizabeth Rothwell, Ben Rubin, and Amjad Tuffaha; my thanks to all of you,
and to anyone else whose name I have inadvertently omitted!
P.J.C.
London
April 2007
Preface to the first edition
The axiomatic method is characteristic of modern mathematics. By making our
assumptionsexplicit,wereducetheriskofmakinganerrorinourreasoningbased
vi Preface
onfalseanalogy;andourresultshaveaclearlydefinedareaofapplicabilitywhich
is as wide as possible (any situation in which the axioms hold).
However, switching quickly from the concrete to the abstract makes a heavy
demandonstudents.Theaxiomaticstyleofmathematicsisusuallymetfirstina
coursewithatitlesuchas‘AbstractAlgebra’,‘AlgebraicStructures’,or‘Groups,
Rings and Fields’. Students who are used to factorising a particular integer or
finding the stationary points of a particular curve find it hard to verify that a
set whose elements are subsets of another set satisfies the axioms for a group,
and even harder to get a feel for what such a group really looks like.
For this reason, among others, I have chosen to treat rings before groups,
although they are logically more complicated. Everyone is familiar with the
set of integers, and can see that it satisfies the axioms for a ring. In the early
stages, when one depends on precedent, the integers form a fairly reliable guide.
Also, the abstract factorisation theorems of ring theory lead to proofs of impor-
tant and subtle properties of the integers, such as the Fundamental Theorem
of Arithmetic. Finally, the path to non-trivial applications is shorter from ring
theory than from group theory.
I have been teaching algebra for the whole of my professional career, and
thisbookreflectsthatexperience.Mostimmediately,itgrewoutoftheAbstract
AlgebracourseatQueenMaryandWestfieldCollege.Chapters2and3arebased
fairly directly on the course content, and provide an introduction to rings (and
fields) and to groups. The first chapter contains essential background material
that every student of mathematics should know, and which can certainly stand
repetition. (A great deal of algebra depends on the concept of an equivalence
relation.)
Chapter 4, on vector spaces, does not try to be a complete account, since
most students would have met vector spaces before they reach this point. The
purpose is twofold: to give an axiomatic approach; and to provide material in
a form which generalises to modules over Euclidean rings, from where two very
importantapplications(finitelygeneratedabeliangroupsandcanonicalformsof
matrices) come.
Chapter 7 carries further the material of Chapters 2 and 3, and also intro-
duces some other types of algebra, chosen for their unifying features: universal
algebra,lattices,andcategories.Thisfollowsachapterinwhichthenumbersys-
tems are defined (so that our earlier trust that the integers form a ring can be
firmly founded), the distinction between algebraic and transcendental numbers
is established, and certain ruler-and-compass construction problems are shown
to be impossible. The final chapter treats two important applications, drawing
on much of what has gone before: coding theory and Galois Theory.
Asmentionedearlier,Chapters2and3canformthebasisofafirstcourseon
algebra,followedbyacoursebasedonChapters5and7.Alternatively,Chapter3
andSections7.1–7.8couldformagrouptheorycourse,andChapters2and5and
Sections7.9–7.14aringtheorycourse.Sections2.14–2.16,6.6–6.8,7.15–7.18,and
8.6–8.11 make up a Galois Theory course. Sections 6.1–6.5, and 6.9–6.10 could
Preface vii
supplement a course on set theory, and Sections 2.14–2.16, 7.15–7.18, and 8.1–
8.5 could be used in conjunction with some material on information theory for
a coding theory course.
Some parts of the book (Sections 7.8, 7.13, and probably the last part of
Chapter 7) are really too sketchy to be used for teaching a course; they are
designed to tempt students into further exploration.
At the end, there is a list of books for further reading, and solutions to
selected exercises from the first three chapters.
Asterisks denote harder exercises.
There is a World Wide Web site associated with this book. It contains solu-
tionstotheremainingexercises,furthertopics,problems,andlinkstoothersites
of interest to algebraists. The address is
http://www.maths.qmul.ac.uk/˜pjc/algebra/
Thanks are due to many generations of students, whose questions and per-
plexities have helped me clarify my ideas and so resulted in a better book than
I might otherwise have written.
P.J.C.
London
December 1997
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Contents
1. Introduction 1
What is mathematics? 1
Numbers 10
Elementary algebra 23
Sets 32
Modular Arithmetic 48
Matrices 52
Appendix: Logic 58
2. Rings 63
Rings and subrings 63
Homomorphisms and ideals 74
Factorisation 87
Fields 99
Appendix: Miscellany 103
3. Groups 110
Groups and subgroups 110
Subgroups and cosets 119
Homomorphisms and normal subgroups 124
Some special groups 132
Appendix: How many groups? 147
4. Vector spaces 150
Vector spaces and subspaces 150
Linear transformations and matrices 161
5. Modules 182
Introduction 182
Modules over a Euclidean domain 190
Applications 196
6. The number systems 209
To the complex numbers 210
Algebraic and transcendental numbers 220
More about sets 230
7. Further topics 237
Further group theory 237
Further ring theory 256